Question

The current through a channel in a fjord is driven by changes in sea level in the ocean outside, η1, caused by tide. The equation for η₁(t) is:

η₁ = η_M2sin(ω_M2t) (eq. 1).

The changes in the water level in the fjord, η₂ (t), depend on the area of the cross-section of the channel, A_C,
stream velocity in the channel, U_C, and surface area of the fjord, A₂. Note that U_C > 0 means that water flows out of the fjord and that U_C < 0 means that water flows into the fjord.
The differential equation describing this is:

(dη₂/dt) = − (A_CU_C)/(A₂) (eq. 2).

The changes in the flow velocity in the channel depend on the differences in water levels on the inside and
outside of the strait and the length of the strait, L_C. The differential equation describing this is:

(dU_C/dt) = g (η₂ - η₁)/L_C (eq. 3).

From eq. 1, 2 and 3 it is possible to make a differential equation for U_C on the form:

U_C'' + aU_c = f(t) (eq. 4)

Show this and decide a and f(t) in the equation over (eq. 4).

ω_M2 = Half-day tide frequency
η_M2 = Half-day tidal amplitude
g = gravity

Answer 1